Finding a Minimum Spanning Tree with a Small Non-Terminal Set

TL;DR

This paper investigates the computational complexity of finding a minimum weight spanning tree with a specified subset of vertices as internal nodes, providing fixed-parameter algorithms and kernelization results.

Contribution

It introduces fixed-parameter algorithms and kernelization techniques for the NP-hard Minimum Weight Non-Terminal Spanning Tree problem, extending known tractability results.

Findings

Provides a $3k$-vertex kernel for the problem.

Develops an $O^*(2^k)$-time algorithm parameterized by non-terminal set size.

Shows fixed-parameter tractability when parameterized by the number of edges in the non-terminal induced subgraph.

Abstract

In this paper, we study the problem of finding a minimum weight spanning tree that contains each vertex in a given subset $V_{\rm NT}$ of vertices as an internal vertex. This problem, called Minimum Weight Non-Terminal Spanning Tree, includes $s$-$t$ Hamiltonian Path as a special case, and hence it is NP-hard. In this paper, we first observe that Non-Terminal Spanning Tree, the unweighted counterpart of Minimum Weight Non-Terminal Spanning Tree, is already NP-hard on some special graph classes. Moreover, it is W[1]-hard when parameterized by clique-width. In contrast, we give a $3k$-vertex kernel and $O^*(2^k)$-time algorithm, where $k$ is the size of non-terminal set $V_{\rm NT}$. The latter algorithm can be extended to Minimum Weight Non-Terminal Spanning Tree with the restriction that each edge has a polynomially bounded integral weight. We also show that Minimum Weight Non-Terminal Spanning Tree is fixed-parameter tractable parameterized by the number of edges in the subgraph induced by the non-terminal set $V_{\rm NT}$, extending the fixed-parameter tractability of Minimum Weight Non-Terminal Spanning Tree to the general case. Finally, we give several results for structural parameterization.